Finite Group Databasebeta

← #349956

Subgroups — #349956

Order 512.

1055 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1055 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2#33148C212yesmaximal
3#25722C211yesmaximal, characteristic
4#9066C214yesmaximal, complemented
5#8928C214yesmaximal, complemented
6#8706C212yesmaximal
7#8342C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#494C2212yes—
10C2×#451—22no—
11#2857C2211yescharacteristic
12#2369—24no—
13#2369—22no—
14#2369C2214yes—
15C2×#391—22no—
16C2×#389—22no—
17C2×C4×C4⋊C4—22no—
18C2×C4×C4⋊C4C2212yes—
19#2016—22no—
20#1987C2214yes—
21#1956C2212yes—
22#1948C2214yes—
23#1920—24no—
24#1853—22no—
25#1823—22no—
26#1797C2212yes—
27C4×C22⋊C8C2211yescharacteristic
28C2×#333C2214yes—
29#1279C2218yes—
30C23×Q8D412yes—
31C22×Q16—24no—
32C2×C4⋊Q8—28no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8D412yes—
35C2×C4⋊Q8C2311yescharacteristic
36C2×C4⋊Q8C2312yes—
37C2×C22⋊Q8—21no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4D412yes—
41C22×C4⋊C4D412yes—
42C22×C42—22no—
43C22×C42—22no—
44C22×C42C2311yescharacteristic
45#494—28no—
46#451—48no—
47C2×#106—24no—
48C2×C4⋊C8C2314yes—
49C2×#103—24no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—22no—
52C2×C4×C8C2312yes—
53#391—28no—
54#391—28no—
55#391—21no—
56#389—28no—
57#389—28no—
58#389—22no—
59#384C2311yescharacteristic
60#380—22no—
61C4×C4⋊C4—28no—
62C4×C4⋊C4—28no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4D414yes—
66C4×C4⋊C4C2314yes—
67C2×#95—24no—
68C2×#95—22no—
69#340—24no—
70#340—24no—
71#340—24no—
72#336—22no—
73#336—21no—
74#333—28no—
75C22×Q8—28no—
76C22×Q8—22no—
77C22×Q8—21no—
78C22×Q8—44no—
79C22×Q8—82no—
80C22×Q8C2×D414yes—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D412yes—
86C2×Q16—48no—
87C2×Q16—48no—
88C2×Q16—84no—
89C22×C8—22no—
90C22×C8—24no—
91C4⋊Q8—216no—
92C4⋊Q8—28no—
93C4⋊Q8—24no—
94C4⋊Q8—24no—
95C4⋊Q8#4118yes—
96C22⋊Q8—44no—
97C2×C4⋊C4—216no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—28no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—84no—
111C2×C4⋊C4C2×D414yes—
112C2×C4⋊C4C2×D414yes—
113C2×C22⋊C4—41no—
114C2×C42—24no—
115C2×C42—24no—
116C2×C42—24no—
117C2×C42—28no—
118C2×C42—44no—
119C2×C42—44no—
120C2×C42—84no—
121C2×C42C2×D412yes—
122C2×C42C2×D411yescharacteristic
123C2×C42C2411yescharacteristic
124#106—48no—
125C4⋊C8—28no—
126#103—48no—
127C22⋊C8—24no—
128C22⋊C8—44no—
129C4×C8—24no—
130#95—216no—
131#95—24no—
132#95—24no—
133#95—42no—
134C24#12011yescharacteristic
135C2×Q8—28no—
136C2×Q8—24no—
137C2×Q8—44no—
138C2×Q8—44no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—82no—
143C2×Q8—82no—
144C2×Q8—816no—
145C2×Q8#12318yes—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—28no—
149C22×C4—28no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—41no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4#12011yescharacteristic
166C22×C4#12012yes—
167C22×C4#13514yes—
168C22×C4#13714yes—
169C22×C4C22×D411yescharacteristic
170C22×C4C22×D412yes—
171Q16—816no—
172C2×C8—24no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—44no—
176C2×C8—44no—
177C2×C8—44no—
178C4⋊C4—28no—
179C4⋊C4—216no—
180C4⋊C4—216no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—82no—
189C4⋊C4—88no—
190C4⋊C4#13718yes—
191C22⋊C4—82no—
192C42—22no—
193C42—24no—
194C42—44no—
195C42—44no—
196C42—48no—
197C42—88no—
198C42C2×#4114yes—
199C23—21no—
200C23—22no—
201C23—21no—
202C23#45214yes—
203C23#45312yes—
204C23C2×#12011yescharacteristic
205Q8—48no—
206Q8—84no—
207Q8—84no—
208Q8—88no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—28no—
214C2×C4—28no—
215C2×C4—21no—
216C2×C4—28no—
217C2×C4—28no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—44no—
221C2×C4—416no—
222C2×C4—416no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—88no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4#45214yes—
237C2×C4C2×#12312yes—
238C2×C4#52514yes—
239C2×C4C2×#13512yes—
240C2×C4C2×#13714yes—
241C2×C4C2×#13712yes—
242C8—44no—
243C8—84no—
244C22—24no—
245C22—22no—
246C22—22no—
247C22—24no—
248C22#163414yes—
249C22#284911yescharacteristic
250C22C2×#45212yes—
251C22#300412yes—
252C22C2×#45311yescharacteristic
253C22#301911yescharacteristic
254C4—22no—
255C4—24no—
256C4—48no—
257C4—41no—
258C4—48no—
259C4—84no—
260C4—161no—
261C4—161no—
262C4#310814yes—
263C4#311314yes—
264C2—24no—
265C2C2×#163411yescharacteristic
266C2#1854611yescharacteristic
267C2#1875312yes—
268C2#3299011yescharacteristic
269C2#3300812yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #9066C2semidirect
5. #8928C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#23692
#331482
#257222
#90662
#89282
#87062
#83422